Finite Group Databasebeta

← #5806

Subgroups — #5806

Order 192.

334 conjugacy classes of subgroups; only the 115 normal subgroups are shown.

normal subgroups only

This group has 334 conjugacy classes of subgroups — too many to draw. Only its 115 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×C3×D4C211yesmaximal, characteristic, complemented
2C2×#237C211yesmaximal, characteristic, complemented
3#953C212yesmaximal, complemented
4C2×#217C212yesmaximal, complemented
5#939C212yesmaximal, complemented
6#938C211yesmaximal, characteristic, complemented
7#936C211yesmaximal, characteristic, complemented
8#935C212yesmaximal, complemented
9D4×Dic3C211yesmaximal, characteristic, complemented
10C4×#67C211yesmaximal, characteristic, complemented
11#924C211yesmaximal, characteristic, complemented
12#546—31nomaximal, Sylow
13C24×C3C2212yes—
14C2×C3×D4—24no—
15C2×C3×D4C2212yes—
16C2×C3×D4C2211yescharacteristic
17C2×C3×D4C2211yescharacteristic
18C22×C3×C4C2211yescharacteristic
19C2×#67C2211yescharacteristic
20C2×#67C2212yescomplemented
21C2×#67C2212yescomplemented
22C22×Dic3C2212yescomplemented
23C22×Dic3C2212yescomplemented
24#237—24no—
25C2×C4×D3C2211yescharacteristic, complemented
26C2×Dic6C2211yescharacteristic, complemented
27#217C2212yes—
28#217C2211yescharacteristic
29#217C2212yescomplemented
30#217C2212yescomplemented
31#217C2212yescomplemented
32#217C2212yescomplemented
33(C2×D3)⋊C4C2211yescharacteristic, complemented
34C4⋊Dic3C2211yescharacteristic, complemented
35Dic3⋊C4C2212yescomplemented
36Dic3⋊C4C2211yescharacteristic, complemented
37C4×Dic3C2211yescharacteristic, complemented
38C2×#41—31no—
39C22×D4D311yescharacteristic, complemented
40#124—31no—
41#123—32no—
42C22⋊Q8—31no—
43#121—31no—
44#121—32no—
45#120—32no—
46C4×D4—31no—
47C4×D4—31no—
48C2×C22⋊C4—32no—
49C23×C3—24no—
50C23×C3—24no—
51C23×C3—42no—
52C23×C3C2312yes—
53C23×C3C2312yes—
54C23×C3C2311yescharacteristic
55C22×D3C2311yescharacteristic
56C3×D4—24no—
57C3×D4—22no—
58C3×D4D414yescomplemented
59C2×C3×C4—22no—
60C2×C3×C4C2311yescharacteristic
61C2×C3×C4C2311yescharacteristic
62#67—24no—
63#67—22no—
64#67—42no—
65C2×Dic3—24no—
66C2×Dic3—42no—
67C2×Dic3C2311yescharacteristic
68C2×Dic3C2311yescharacteristic
69C2×Dic3C2312yes—
70C2×Dic3C2312yes—
71C2×Dic3C2311yescharacteristic
72C4×D3—22no—
73Dic6—22no—
74C24C2×D312yes—
75#41—64no—
76C2×Q8—31no—
77C2×D4—24no—
78C2×D4—31no—
79C2×D4—32no—
80C2×D4—32no—
81C2×D4C2×D312yes—
82C2×D4C2×D311yescharacteristic
83C2×D4C2×D311yescharacteristic
84C22×C4—32no—
85C22×C4—32no—
86C22×C4—31no—
87C22×C4C2×D311yescharacteristic
88C4⋊C4—31no—
89C4⋊C4—31no—
90C4⋊C4—32no—
91C22⋊C4—31no—
92C22⋊C4—32no—
93C22⋊C4—32no—
94C22⋊C4—32no—
95C22⋊C4—32no—
96C22⋊C4—32no—
97C22⋊C4—31no—
98C42—31no—
99C22×C3—21no—
100C22×C3—22no—
101C22×C3—22no—
102C22×C3—21no—
103C22×C3—22no—
104C22×C3—22no—
105C22×C3—22no—
106C22×C3—48no—
107C22×C3C2×D414yes—
108C22×C3#4112yescomplemented
109C22×C3C2411yescharacteristic
110C2×D3—21no—
111C2×D3—21no—
112C2×D3—21no—
113C3×C4—21no—
114C3×C4C2×D412yes—
115Dic3—22no—
116Dic3—21no—
117Dic3—21no—
118Dic3—21no—
119Dic3—22no—
120C23—24no—
121C23—24no—
122C23—31no—
123C23—42no—
124C23C22×D312yes—
125C23C22×D311yescharacteristic
126C23C22×D312yes—
127Q8—62no—
128D4—24no—
129D4—22no—
130D4—62no—
131D4—64no—
132D4—122no—
133D4#6714yescomplemented
134C2×C4—22no—
135C2×C4—31no—
136C2×C4—31no—
137C2×C4—32no—
138C2×C4—31no—
139C2×C4—32no—
140C2×C4—64no—
141C2×C4—62no—
142C2×C4—122no—
143C2×C4C22×D311yescharacteristic
144C2×C4C22×D311yescharacteristic
145C2×C3—22no—
146C2×C3—24no—
147C2×C3—42no—
148C2×C3C22×D411yescharacteristic
149C2×C3C2×#4111yescharacteristic
150C2×C3H(5,𝔽2)11yescharacteristic
151D3—41no—
152C22—22no—
153C22—22no—
154C22—21no—
155C22—21no—
156C22—22no—
157C22—22no—
158C22—22no—
159C22—48no—
160C22—61no—
161C22—61no—
162C22—61no—
163C22#23712yescomplemented
164C22C2×#6714yes—
165C22C23×D311yescharacteristic
166C4—21no—
167C4—61no—
168C4—61no—
169C4—62no—
170C4—61no—
171C4—62no—
172C4C2×#6712yes—
173C3#54611yescharacteristic, complemented, Sylow
174C2—22no—
175C2—24no—
176C2—42no—
177C2—121no—
178C2C2×#23711yescharacteristic
179C2#100411yescharacteristic
180C2C22×#6711yescharacteristic
subgroupG/H
center, FrattiniC22C23×D3
commutator, socle, cosocleC22×C3C24
FittingC22×C3×D4C2
hypercenterC22×D4D3
hypocenterC3#546
LCSG ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C22×D4

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
1. C22×C3×D4C2semidirect
2. C2×#237C2semidirect
3. #953C2semidirect
4. C2×#217C2semidirect
5. #939C2semidirect
6. #938C2semidirect
7. #936C2semidirect
8. #935C2semidirect
9. D4×Dic3C2semidirect
10. C4×#67C2semidirect
11. #924C2semidirect
20. C2×#67C22semidirect
21. C2×#67C22semidirect
22. C22×Dic3C22semidirect
23. C22×Dic3C22semidirect
25. C2×C4×D3C22semidirect
26. C2×Dic6C22semidirect
29. #217C22semidirect
30. #217C22semidirect
31. #217C22semidirect
32. #217C22semidirect
33. (C2×D3)⋊C4C22semidirect
34. C4⋊Dic3C22semidirect
35. Dic3⋊C4C22semidirect
36. Dic3⋊C4C22semidirect
37. C4×Dic3C22semidirect
39. C22×D4D3semidirect
58. C3×D4D4semidirect
108. C22×C3#41semidirect
133. D4#67semidirect
163. C22#237semidirect
173. C3#546semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
C22×C3×D42
C2×#2372
#9532
C2×#2172
#9392
#9382
#9362
#9352
D4×Dic32
C4×#672
#9242
#5463